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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfnf 2301 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
𝑥𝜑       𝑥𝑦𝜑
 
Theoremnfnf1OLD 2302 Obsolete proof of nfnf1 2176 as of 12-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑
 
Theoremaxc16gOLD 2303* Obsolete proof of axc16g 2277 as of 11-Oct-2021. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2387, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2282. (Revised by Wolf Lammen, 11-Oct-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
 
Theoremaxc16nfOLD 2304* Obsolete proof of axc16nf 2280 as of 12-Oct-2021. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2179. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
 
Theorem19.12 2305 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 2321 and r19.12sn 4395. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
(∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremnfald 2306 Deduction form of nfal 2296. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) (Proof shortened by Wolf Lammen, 16-Oct-2021.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)
 
TheoremnfaldOLD 2307 Obsolete proof of nfald 2306 as of 16-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)
 
Theoremnfexd 2308 If 𝑥 is not free in 𝜓, it is not free in 𝑦𝜓. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)
 
Theoremnfa2OLD 2309 Obsolete proof of nfa2 2185 as of 18-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑦𝑥𝜑
 
TheoremexanOLDOLD 2310 Obsolete proof of exan 1933 as of 7-Jul-2021. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)
 
Theoremaaan 2311 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
 
Theoremeeor 2312 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
 
Theoremcbv3v 2313* Version of cbv3 2406 with a dv condition, which does not require ax-13 2387. (Contributed by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)
 
Theoremdvelimhw 2314* Proof of dvelimh 2472 without using ax-13 2387 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 23-Dec-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremcbv3hv 2315* Version of cbv3h 2407 with a dv condition on 𝑥, 𝑦, which does not require ax-13 2387. Was used in a proof of axc11n 2447 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.)
(𝜑 → ∀𝑦𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)
 
Theoremcbvalv1 2316* Version of cbval 2412 with a dv condition, which does not require ax-13 2387. See cbvalvw 2116 for a version with two dv conditions, requiring fewer axioms, and cbvalv 2414 for another variant. (Contributed by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
Theoremcbvexv1 2317* Version of cbvex 2413 with a dv condition, which does not require ax-13 2387. See cbvexvw 2117 for a version with two dv conditions, requiring fewer axioms, and cbvexv 2416 for another variant. (Contributed by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
Theoremequs5aALT 2318 Alternate proof of equs5a 2481. Uses ax-12 2192 but not ax-13 2387. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremequs5eALT 2319 Alternate proof of equs5e 2482. Uses ax-12 2192 but not ax-13 2387. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 
Theorempm11.53 2320* Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 2067 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
 
Theorem19.12vv 2321* Special case of 19.12 2305 where its converse holds. See 19.12vvv 2068 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
 
Theoremeean 2322 Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theoremeeanv 2323* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theoremeeeanv 2324* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
 
Theoremee4anv 2325* Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
(∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
 
TheoremcleljustALT 2326* Alternate proof of cleljust 2143. It is kept here and should not be modified because it is referenced on the Metamath Proof Explorer Home Page (mmset.html) as an example of how DV conditions are inherited by substitutions. (Contributed by NM, 28-Jan-2004.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
 
TheoremcleljustALT2 2327* Alternate proof of cleljust 2143. Compared with cleljustALT 2326, it uses nfv 1988 followed by equsexv 2252 instead of ax-5 1984 followed by equsexhv 2267, so it uses the idiom 𝑥𝜑 instead of 𝜑 → ∀𝑥𝜑 to express non-freeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
 
Theoremaxc11r 2328 Same as axc11 2452 but with reversed antecedent. Note the use of ax-12 2192 (and not merely ax12v 2193). (Contributed by NM, 25-Jul-2015.)
(∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
 
TheoremnfrOLD 2329 Obsolete proof of nf5r 2207 as of 6-Oct-2021. (Contributed by Mario Carneiro, 26-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
TheoremnfriOLD 2330 Obsolete proof of nf5ri 2208 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (𝜑 → ∀𝑥𝜑)
 
TheoremnfrdOLD 2331 Obsolete proof of nf5rd 2209 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))
 
TheoremalimdOLD 2332 Obsolete proof of alimd 2224 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
TheoremalrimiOLD 2333 Obsolete proof of alrimi 2225 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)
 
TheoremnfdOLD 2334 Obsolete proof of nf5d 2261 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
TheoremnfdhOLD 2335 Obsolete proof of nf5dh 2171 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
TheoremalrimddOLD 2336 Obsolete proof of alrimdd 2226 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
TheoremalrimdOLD 2337 Obsolete proof of alrimd 2227 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
TheoremeximdOLD 2338 Obsolete proof of eximd 2228 as of 6-Oct-2021. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
TheoremnexdOLD 2339 Obsolete proof of nexd 2232 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)
 
TheoremalbidOLD 2340 Obsolete proof of albid 2233 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 
TheoremexbidOLD 2341 Obsolete proof of exbid 2234 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
TheoremnfbidfOLD 2342 Obsolete proof of nfbidf 2235 as of 6-Oct-2021. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
 
Theorem19.3OLD 2343 Obsolete proof of 19.3 2212 as of 6-Oct-2021. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (∀𝑥𝜑𝜑)
 
Theorem19.9dOLD 2344 Obsolete proof of 19.9d 2213 as of 6-Oct-2021. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))
 
Theorem19.9tOLD 2345 Obsolete proof of 19.9t 2214 as of 6-Oct-2021. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof shortened by Wolf Lammen, 14-Jul-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
 
Theorem19.9OLD 2346 Obsolete proof of 19.9 2215 as of 6-Oct-2021. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (∃𝑥𝜑𝜑)
 
Theorem19.9hOLD 2347 Obsolete proof of 19.9h 2263 as of 6-Oct-2021. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) (Proof shortened by Wolf Lammen, 14-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∃𝑥𝜑𝜑)
 
Theoremnfa1OLDOLD 2348 Obsolete proof of nfa1 2173 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑
 
Theoremnfnf1OLDOLD 2349 Obsolete proof of nfnf1 2176 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑
 
TheoremnfntOLD 2350 Obsolete proof of nfnt 1927 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
 
TheoremnfnOLD 2351 Obsolete proof of nfn 1929 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       𝑥 ¬ 𝜑
 
TheoremnfndOLD 2352 Obsolete proof of nfnd 1930 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)
 
Theorem19.21t-1OLD 2353 One direction of the bi-conditional in 19.21t 2216. Unlike the reverse implication, it does not depend on ax-10 2164. Obsolete as of 6-Oct-2021 (Contributed by Wolf Lammen, 4-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
Theorem19.21tOLD 2354 Obsolete proof of 19.21t 2216 as of 6-Oct-2021. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 
Theorem19.21OLD 2355 Obsolete proof of 19.21 2218 as of 6-Oct-2021. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
 
Theorem19.21-2OLD 2356 Obsolete proof of 19.21-2 2221 as of 6-Oct-2021. (Contributed by NM, 4-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
 
Theorem19.21hOLD 2357 Obsolete proof of 19.21h 2264 as of 6-Oct-2021. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
 
Theoremstdpc5OLDOLD 2358 Obsolete proof of stdpc5 2219 as of 6-Oct-2021. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2164. (Revised by Wolf Lammen, 4-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
 
Theorem19.23tOLD 2359 Obsolete proof of 19.23t 2222 as of 6-Oct-2021. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
 
Theorem19.23OLD 2360 Obsolete proof of 19.23 2223 as of 6-Oct-2021. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.23hOLD 2361 Obsolete proof of 19.23h 2265 as of 6-Oct-2021. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
TheoremexlimiOLD 2362 Obsolete proof of exlimi 2229 as of 6-Oct-2021. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)
 
TheoremexlimihOLD 2363 Obsolete proof of exlimih 2291 as of 6-Oct-2021. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)
 
TheoremexlimdOLD 2364 Obsolete proof of exlimd 2230 as of 6-Oct-2021. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))
 
TheoremexlimdhOLD 2365 Obsolete proof of exlimdh 2292 as of 6-Oct-2021. (Contributed by NM, 28-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))
 
TheoremnfdiOLD 2366 Obsolete proof of nf5di 2262 as of 6-Oct-2021. (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜑)       𝑥𝜑
 
TheoremnfimdOLD 2367 Obsolete proof of nfimd 1968 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
Theoremhbim1OLD 2368 Obsolete proof of hbim 2270 as of 6-Oct-2021. (Contributed by NM, 2-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theoremnfim1OLD 2369 Obsolete proof of nfim1 2210 as of 6-Oct-2021. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)
 
TheoremnfimOLD 2370 Obsolete proof of nfim 1970 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
TheoremhbimdOLD 2371 Obsolete proof of hbimd 2269 as of 6-Oct-2021. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))
 
TheoremhbimOLD 2372 Obsolete proof of hbim 2270 as of 6-Oct-2021. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
TheoremnfandOLD 2373 Obsolete proof of nfand 1971 as of 6-Oct-2021. (Contributed by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
Theoremnf3andOLD 2374 Obsolete proof of nf3and 1972 as of 6-Oct-2021. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑥𝜃)       (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))
 
Theorem19.27OLD 2375 Obsolete proof of 19.27 2238 as of 6-Oct-2021. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 
Theorem19.28OLD 2376 Obsolete proof of 19.28 2239 as of 6-Oct-2021. (Contributed by NM, 1-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
 
Theoremnfan1OLD 2377 Obsolete proof of nfan1 2211 as of 6-Oct-2021. (Contributed by Mario Carneiro, 3-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)
 
TheoremnfanOLDOLD 2378 Obsolete proof of nfan 1973 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
TheoremnfnanOLD 2379 Obsolete proof of nfnan 1975 as of 6-Oct-2021. (Contributed by Scott Fenton, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnf3anOLD 2380 Obsolete proof of nf3an 1976 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)
 
TheoremhbanOLD 2381 Obsolete proof of hban 2271 as of 6-Oct-2021. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theoremhb3anOLD 2382 Obsolete proof of hb3an 2272 as of 6-Oct-2021. (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
 
TheoremnfbidOLD 2383 Obsolete proof of nfbid 1977 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
TheoremnfbiOLD 2384 Obsolete proof of nfbi 1978 as of 6-Oct-2021. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
TheoremnforOLD 2385 Obsolete proof of nfor 1979 as of 6-Oct-2021. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnf3orOLD 2386 Obsolete proof of nf3or 1980 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)
 
1.5.4  Axiom scheme ax-13 (Quantified Equality)
 
Axiomax-13 2387 Axiom of Quantified Equality. One of the equality and substitution axioms of predicate calculus with equality.

An equivalent way to express this axiom that may be easier to understand is 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧))) (see ax13b 2111). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent ¬ 𝑥 = 𝑦 to hold, 𝑥 and 𝑦 must have different values and thus cannot be the same object-language variable (so they are effectively "distinct variables" even though no $d is present). Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-5 1984, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows. Note that ax-5 1984 cannot prove this because its distinct variable ($d) requirement is not satisfied directly but only indirectly (outside of Metamath) by the argument above.

The original version of this axiom was ax-c9 34675 and was replaced with this shorter ax-13 2387 in December 2015. The old axiom is proved from this one as theorem axc9 2443.

The primary purpose of this axiom is to provide a way to introduce the quantifier 𝑥 on 𝑦 = 𝑧 even when 𝑥 and 𝑦 are substituted with the same variable. In this case, the first antecedent becomes ¬ 𝑥 = 𝑥 and the axiom still holds.

Although this version is shorter, the original version axc9 2443 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2443 is in dvelimh 2472 which converts a distinct variable pair to the distinctor antecedent ¬ ∀𝑥𝑥 = 𝑦. In particular, it is conjectured that it is not possible to prove ax6 2392 from ax6v 2051 without this axiom.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. To show that, we add these restrictions to theorem ax13v 2388 and use only ax13v 2388 for further derivations. Thus, ax13v 2388 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2388 (preferred) or ax13 2390 (if the stronger form is needed).

This axiom scheme is logically redundant (see ax13w 2158) but is used as an auxiliary axiom scheme to achieve scheme completeness (i.e. so that all possible cases of bundling can be proved; see text linked at mmtheorems.html#ax6dgen). It is not known whether this axiom can be derived from the others. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theoremax13v 2388* A weaker version of ax-13 2387 with distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. In order to show (with ax13 2390) that this weakening is still adequate, this should be the only theorem referencing ax-13 2387 directly.

Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1984. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. (Contributed by NM, 30-Jun-2016.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theoremax13lem1 2389* A version of ax13v 2388 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2390 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 
Theoremax13 2390 Derive ax-13 2387 from ax13v 2388 and Tarski's FOL. This shows that the weakening in ax13v 2388 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage (Revised by Wolf Lammen, 2-Jun-2021.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theoremax6e 2391 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-4 1882 through ax-9 2144, all axioms other than ax-6 2050 are believed to be theorems of free logic, although the system without ax-6 2050 is not complete in free logic.

It is preferred to use ax6ev 2052 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2389 became available. (Revised by Wolf Lammen, 8-Sep-2018.)

𝑥 𝑥 = 𝑦
 
Theoremax6 2392 Theorem showing that ax-6 2050 follows from the weaker version ax6v 2051. (Even though this theorem depends on ax-6 2050, all references of ax-6 2050 are made via ax6v 2051. An earlier version stated ax6v 2051 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 2050 so that all proofs can be traced back to ax6v 2051. When possible, use the weaker ax6v 2051 rather than ax6 2392 since the ax6v 2051 derivation is much shorter and requires fewer axioms. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Theoremaxc10 2393 Show that the original axiom ax-c10 34671 can be derived from ax6 2392 and axc7 2275 (on top of propositional calculus, ax-gen 1867, and ax-4 1882). See ax6fromc10 34681 for the rederivation of ax6 2392 from ax-c10 34671.

Normally, axc10 2393 should be used rather than ax-c10 34671, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 
Theoremspimt 2394 Closed theorem form of spim 2395. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
 
Theoremspim 2395 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2395 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspimed 2396 Deduction version of spime 2397. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))
 
Theoremspime 2397 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theoremspimv 2398* A version of spim 2395 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2258 and spimvw 2078. See also spimvALT 2399. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2164. (Revised by BJ, 29-Nov-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
TheoremspimvALT 2399* Alternate proof of spimv 2398. Shorter but requires more axioms. (Contributed by NM, 31-Jul-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspimev 2400* Distinct-variable version of spime 2397. (Contributed by NM, 10-Jan-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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